**Memorandum 1806**

*Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation*

O.V. Nechaev, E.P. Shurina & M.A. Botchev

**Abstract:**

In the edge vector finite element solution of the frequency
domain Maxwell equations,
the presence of a large kernel of the discrete rotor operator
is known to ruin convergence of standard iterative solvers.
We extend the approach of [1] and, using
domain decomposition ideas, construct a
multilevel iterative solver where the projection with respect
to the kernel is combined with the use of a hierarchical
representation of the vector finite elements.

The new iterative scheme appears to be an efficient solver for the
edge finite element solution of the frequency domain Maxwell equations.
The solver can be seen as a variable preconditioner
and, thus, accelerated by Krylov
subspace techniques (e.g. GCR or FGMRES).
We demonstrate the efficiency of our approach on a test
problem with strong jumps in the conductivity.

[1] R. Hiptmair.
Multigrid method for Maxwell's equations.
*SIAM J. Numer. Anal.*, 36(1):204-225, 1999.

**Keywords:**
Nédélec vector finite elements; kernel of the rotor operator;
multilevel iterative solvers; hierarchical preconditioners;
domain decomposition.

**Mathematics Subject Classification:**
65N22, 65N30, 65N55

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